Semiconductor quantum dots or ‘artificial atoms’ provide highly tunable structures for trapping and manipulating individual electrons9,10,11. Such quantum dots are promising candidates as qubits for quantum computation12,13,14, owing in part to the long lifetimes and slow dephasing of electron spins in semiconductors7,15. Si quantum dots are predicted to have especially long lifetimes and slow dephasing, due to low spin–orbit interaction and low nuclear spin density16,17. In the past several years, much activity has focused on the development of quantum dots in Si/SiGe (refs 18, 19, 20, 21, 22) and recent advances in materials quality and fabrication techniques have enabled the observation of coherent spin phenomena in such quantum dots23.

Spin-to-charge conversion, in which spin states are detected through their effect on charge motion, enables measurement of individual electron spins in quantum dots15. Spin blockade is the canonical example of spin-to-charge conversion in transport, where charge current is blocked in a double quantum dot by a metastable spin state. The blockade occurs when one electron is confined in the left dot and a further electron enters the right dot forming a spin triplet state T(1,1) (Fig. 1a). Exiting the dot requires reaching the triplet T(2,0), with both electrons in the left dot, a state that is higher in energy. The electron is thus trapped in the right dot, unless relaxation from T(1,1) to S(1,1) occurs, opening a downhill channel through S(2,0). As we show below, this aspect of spin blockade in Si is virtually identical to that previously observed in other systems1,2,3.

Figure 1: Pauli spin blockade regime in a double quantum dot.
figure 1

a, Energy level schematic diagram of a double quantum dot with one electron confined in the left dot (black). The applied bias causes electron flow from the right lead (R) to the left (L). Incoming electrons can form either a spin singlet S(1,1) (green) or a spin triplet T(1,1) (red). Electrons forming a singlet have an accessible fast channel through S(2,0) to the left lead. In contrast, electrons entering the triplet T(1,1) cannot exit through T(2,0), resulting in metastable occupation of the T(1,1) state, and causing spin blockade of the current. b, The identical energy level configuration as in a, but with the direction of current flow reversed. Electrons entering S(2,0) have no fast path through the system. If S(2,0) loads more rapidly than it unloads, current will be blockaded. In contrast, electrons entering T(2,0) have an accessible fast channel through T(1,1), provided spin relaxation to S(2,0) (blue wavy arrow) does not occur. Electron transport through the triplets, supported by a long spin lifetime, is denoted LET.

The unexpected effect presented here is lifetime-enhanced transport (LET). The energy level diagram for LET is the same as for spin blockade, except that current flows in the opposite direction (Fig. 1b). Flow through the triplet channel is now energetically downhill, whereas flow through the singlet channel is very slow, because it requires either an uphill transition or tunnelling directly from the left dot to the right lead. Transport current will be observable only if electrons flow almost exclusively through the triplet channel, requiring even slower triplet–singlet relaxation rates than those needed to observe spin blockade.

The tunable quantum dot used in these experiments was formed in a Si/SiGe heterostructure. The gate structure (Fig. 2a) has the shape often associated with a single quantum dot, and the corresponding Coulomb diamonds are shown in Fig. 2b. By tuning the gate voltages, the single dot was split into two tunnel-coupled quantum dots. Such transformations of a lateral single quantum dot into multiple quantum dots have been demonstrated in similar systems24,25. Here, by changing voltages on gates G and CS, and keeping those on BL, T and BR fixed, the electron occupations are tuned while keeping the tunnel barriers constant (Fig. 2d). The left dot is coupled more strongly to gate G, and the right dot is coupled more strongly to gate CS. The electron occupancies indicated in the figure correspond to an equivalent charge configuration with a single unpaired spin in the (1,0) state.

Figure 2: Formation of a double quantum dot.
figure 2

a, False-colour micrograph of a device similar to the one used in the experiments. A 2DEG was formed in a 12 nm strained silicon quantum well with a sheet carrier density of 4×1011 cm−2 and a mobility of 40,000 cm2 V−1 s−1. Ohmic contacts (indicated schematically by red squares) were formed by annealing an alloy of Au:Sb(1%) at 550 C. Metal gates used to form the quantum dots were realized by depositing palladium on the sample surface and are labelled on the micrograph. The white arrow indicates the direction of electron flow when VSD>0. b, Magnitude of the measured current as a function of source–drain voltage VSD and the voltage on gate G. The black regions indicate Coulomb blockade where the number of electrons in the dot is fixed. Outside this blockade, single-electron tunnelling through the dot occurs. c, A numerical simulation of the charge density for the gates as shown and for gate voltages corresponding to the double quantum dot data (d). Two electrons prefer to occupy opposite sides of the open region between the gates. d, The single quantum dot was deformed into two tunnel-coupled dots in series by using a combination of negative voltages on gates T, BL and BR. The magnitude of the measured current through the double quantum dot is plotted as a function of the voltages on gates G and CS, with VSD=0.1 mV. The dot coupled more to gate G (CS) is the left (right) dot. As described in the text, the notation (m,n) represents the effective left and right dot electron occupancy, and the triple point studied in detail here appears inside the blue circle.

The region of interest here is indicated by the blue dashed circle in Fig. 2d. The lower of these ‘triple points’ corresponds to degeneracy between the (1,0), (1,1) and (2,0) charge states26,27. In this regime, an electron with a spin |↑〉 or |↓〉 is confined in the left dot, and the incoming electron can form either a spin singlet or any of the spin triplets T+:|↑↑〉, T:|↓↓〉 or , which are degenerate at zero magnetic field. The singlet–triplet energy splitting is larger for two electrons occupying the same dot (2,0), than when they are in different dots (1,1), resulting in the energy level schematic diagrams shown in Fig. 1.

Spin blockade arises because spin is conserved during tunnelling, preventing the direct transition from the triplet T(1,1) to the singlet S(2,0). We observe this blockade as shown in Fig. 3a–c. These measurements are taken at finite bias, where the triple points expand into bias triangles28. When T(1,1) is loaded and no relaxation occurs from T(1,1) to S(1,1), spin blockade is observed (as marked by the orange triangle) and the bias triangle is truncated as shown schematically in Fig. 3b. The observed current in the blockaded region is limited by the noise floor in the measurement (7 fA r.m.s.). Spin blockade is fully lifted when the T(2,0) state is brought below the T(1,1) state (blue star).

Figure 3: Spin blockade and LET.
figure 3

a, Positive bias (VSD=0.2 mV) triangles representing the (1,0)–(1,1)–(2,0)–(2,1) charge transition. Current through the device is blocked owing to Pauli spin blockade (as marked by the orange triangle) in the region outlined by yellow dashed lines. The blockade is lifted when the triplet state becomes accessible (blue star). The lower triangle is referred to as the ‘electron triangle’ and the upper triangle is called the ‘hole triangle’. The blue arrows represent the energy axis Σ where the energy levels of both dots are changed together, and the detuning axis Δ where dot levels are moved in opposite directions. Note that the edges of the spin blockade regime (green circle) show measurable current flow due spin exchange with the leads2. b, Schematic representation of the positive bias triangles. Dashed lines mark the blockaded region, whereas filled yellow regions indicate no blockade. The energy gap between the spin singlet and triplet (EST) is indicated along the edge of the triangles. c, Transport details and energy level schematic diagrams for the electron triangle corresponding to the points denoted by the green circle, orange triangle and blue star in a and b. The sigma directions in a and b correspond to moving the levels on the left and right up and down together, and the delta directions correspond to moving the levels on the left and the right in opposite directions. The same pattern holds for parts df. d, Negative bias (VSD=−0.3 mV) triangles through the same charge transition. Current flow through full electron and hole triangles is observed (white dashed lines are shown on the electron triangle). In addition, strong ‘tails’ of current (green dashed parallelograms) are observed at the base of the triangles. These extensions arise due to LET, as discussed in the text. e, Schematic representation of the negative bias triangles. Regions outlined by the solid back lines correspond to conventional transport, whereas those outlined by solid green lines (the tails) correspond to LET. Lengths corresponding to EST are labelled. f, Transport details and energy level schematic diagrams for the electron triangle corresponding to the points denoted by the red square, purple diamond and teal cross in d and e.

Now consider the same energy level configuration, but with opposite bias across the dot (Fig. 1b). In previous work, this configuration has been shown to be blockaded2,8. In contrast, here in Si we observe a strong ‘tail’ of current in this configuration, corresponding to the extra parallelograms (green outline) in Fig. 3d,e. As shown in detail below, the condition for observing this tail is that the metastable S(2,0) state must be loaded more slowly than it empties. The relaxation rate from the T(2,0) state into S(2,0) sets a lower bound for this loading rate. Because the measured current at the point labelled (+) is significant only when the spin lifetime of T(2,0) is long, we denote this tail of current the triplet tail and the effect LET.

The dimensions of the triplet tail in the charge stability diagram (Fig. 3d,e) provide a measurement of the energy difference between the (2,0) triplet and singlet states (EST=ETES). Both the length of the tail and the distance between the tail and the edge of the bias triangle correspond to EST (Fig. 3e). This (2,0) singlet–triplet energy gap as extracted from the data is 240±30 μeV.

A simple rate model gives insight into when LET occurs. The rates in the model correspond to transitions between the states shown in Fig. 1b, and the corresponding lifetimes are the inverses of the rates. By calculating the expected amount of time required for an electron to pass through the system, we obtain a quantity proportional to the measured current I (see Supplementary Information for the complete analysis). The slow rates are of interest here: the relaxation rate ΓTS from the triplet T(2,0) to the singlet S(2,0), the loading rate ΓLS of the singlet S(2,0) from the lead and the unloading rate ΓS of the singlet S(2,0). To focus on these rates and to develop intuition, we assume that all other rates are equal to a single rate, Γfast, an assumption that does not change the qualitative understanding. The resulting proportionality for the current is

As this proportionality shows, the triplet tail is observed if and only if the sum of the triplet–singlet relaxation rate ΓTS and the loading rate from the lead ΓLS is not large compared with the escape rate ΓS. If the triplet–singlet relaxation rate is much faster than the escape rate, then the tail regime will be blockaded by electrons trapped in the S(2,0) state. In our experiments, essentially no reduction in current (5%) is observed moving from the bias triangle into the tail (from the blue diamond towards the teal cross in Fig. 3d). Thus, electrons are rarely trapped in S(2,0), indicating that the triplet–singlet relaxation rate ΓTS and the loading rate from the left lead ΓLS are both much less than ΓS, itself a slow rate. A similar calculation with the opposite bias shows that the condition for spin blockade is that the triplet–singlet relaxation rate in the (1,1) configuration is much slower than the fast rates, a far less stringent condition.

To investigate the rapid tunnelling between dots 1 and 2, and to understand the device physics, we have modelled the device numerically, as shown in Fig. 2c. Established methods are used to treat the various charge regions self-consistently, including trapped surface charge, ionized dopants, the two-dimensional electron gas (2DEG) and the device29. The dopants are treated in the jellium approximation, whereas the inhomogeneous depletion of the 2DEG is treated semiclassically in a 2D Thomas–Fermi approximation. For the gated region, a 2D Hartree–Fock basis of single-electron orbitals is obtained from the effective mass envelope equation, and a two-electron singlet wavefunction is constructed using the configuration interaction method, similar to ref. 14. The results show that the bottom of the quantum dot confinement potential is nearly flat, with an oblong shape about 200 nm across. General arguments suggest that the electron–electron interactions should dominate the kinetic energy in silicon for electrons separated by over 100 nm, causing two electrons to form a double dot. The modelling results, shown in Fig. 2c, confirm that these general arguments give the correct intuition. As is clear from the figure, the effective tunnel barrier between the two dots is low, consistent with LET. We note that quantum dot splitting has been observed elsewhere, where it was attributed to deformation by a gate potential24 or a local impurity25. Although inhomogeneous confinement may also be present in our device, it is not needed to explain the double dot.

LET should be observable in many materials systems, provided the appropriate ratio of rates can be obtained. Indeed, slow triplet–singlet relaxation and the preferential loading of triplets versus singlets have both been observed in GaAs quantum dots, in pulsed-gate experiments30. By analysing the current versus voltage data in our bias triangles (see the Supplementary Information), we find that in the tail regime, triplet loading occurs at a rate at least 1,000 times greater than singlet loading. This ratio is 50 times greater than in previous observations of spin-dependent tunnelling30, which may lead to corresponding enhancements in spin readout. The singlet loading is suppressed because its tunnel barrier to the external lead is larger than that of the triplet state31; the relatively large effective mass of Si enhances this suppression. The higher unloading rate is a consequence of the relatively small tunnel barrier between the two dots, as confirmed by numerical modelling. Our bound on the singlet loading rate places a weak bound on triplet–singlet relaxation of ΓTS<63,000 s−1, although the actual value is expected to be much smaller16.

The phenomena described above of spin blockade and its complementary effect of LET can be unified by measurements of the system with an applied inplane magnetic field. In a magnetic field, the spin triplets are split linearly by the Zeeman energy (EZ=g μBB SZ), where μB is the Bohr magneton, B is the magnetic field and SZ is +1 for |↑↑〉, −1 for |↓↓〉 and 0 for . As the g-factor is positive for silicon, the T states shift lower in energy compared with the T0 states, providing a technique for testing the interpretation of the data proposed above.

Figure 4a–c (d–f) shows the energy diagrams for the cases where EZ is less (more) than EST. In Fig. 4a–c, the ground state of the (1,1) configuration is T(1,1), whereas that of the (2,0) configuration is S(2,0). Spin blockade now occurs in a smaller region than at B=0, as indicated by dashed lines and the orange circle in Fig. 4a. Spin blockade is lifted in the conventional way when T(2,0) is lowered below T(1,1), corresponding to the triangular regions on the lower right in Fig. 4a,b. Spin blockade is also lifted when the S(1,1) state can participate in transport (blue diamond). However, the lifting of the blockade in this case is due to LET, because this S(1,1) state is an excited state of the (1,1) configuration, giving rise to a singlet tail in Fig. 4a,b. This tail is a striking example of a generalization of LET: the singlet–triplet splitting is now inverted, and the LET is now due to long lifetimes in the singlet channel rather than the triplet channel. LET can be generalized to any situation where electron transport occurs through long-lived excited states, whereas lower energy states that would be metastably trapped are avoided.

Figure 4: Zeeman splitting, spin blockade and LET.
figure 4

a, Magnitude of the measured current with an inplane magnetic field of 1.5 T applied such that EZ is less than EST. Spin blockade is reduced to a smaller region indicated by dashed yellow lines. Note that there is no conduction along the edges because the S(1,1) state is no longer energetically accessible when the (1,1) ground state T(1,1) aligns with the lead, preventing spin exchange. In the region labelled with a blue diamond, a tail appears corresponding to transport through the excited spin singlet state. b, Schematic illustration of the bias triangles for small inplane magnetic field. The energies EZ and EST−=ESTEZ can be extracted from the graph as indicated by red and blue arrows. The angles in this schematic diagram are exaggerated relative to a. c, Energy level schematic diagrams at the points labelled by the orange circle and blue diamond in a and b. d, Magnitude of the measured current with an inplane magnetic field of 3.5 T applied such that EZ is larger than EST. No blockade is observed inside the triangles. Excited states are observed inside the triangles as bright lines parallel to the triangle bases. The tail regions correspond to LET through excited spin singlet states. e, Schematic illustration of the bias triangles for large inplane magnetic field. The energy EZ can be extracted from the graph as indicated by the red arrow. The angles in this schematic diagram are exaggerated relative to d. f, Energy level schematic diagrams at the points labelled by the purple diamond and green star in d and e.

When EZ>EST (Fig. 4d–f), the ground state configurations are T(1,1) and T(2,0), and there should be no spin blockade because ground-state transitions are allowed (purple diamond). Our measurements indeed show full triangles with no blockade. From the features visible inside the triangles (bright lines parallel to base), various excited states can be identified. LET also occurs through excited spin singlet states in this configuration, giving rise to a tail (green star). These data demonstrate the existence of long-lived electron spin states even in the presence of a finite magnetic field, a requirement for various quantum operations.